machines

Article

ADRC-Based Robust and Resilient Control of a5-Phase PMSM Driven Electric Vehicle

Abir Hezzi 1, Seifeddine Ben Elghali 2 , Yemna Bensalem 1, Zhibin Zhou 3,Mohamed Benbouzid 4,5,* and Mohamed Naceur Abdelkrim 1

1 MACS LR16ES22, University of Gabes, Gabes 6072, Tunisia; abir.hezzi@gmail.com (A.H.);bensalem.yemna2017@gmail.com (Y.B.); naceur.abdelkrim@enig.rnu.tn (M.N.A.)

2 Laboratoire d’Informatique & Systèmes (UMR CNRS 7020), University of Aix-Marseille,13397 Marseille, France; seifeddine.benelghali@lis-lab.fr

3 Institut de Recherche Dupuy de Lôme (UMR CNRS 6027), ISEN Yncréa Ouest Brest, 29200 Brest, France;zhibin.zhou@isen-ouest.yncrea.fr

4 Institut de Recherche Dupuy de Lôme (UMR CNRS 6027), University of Brest, 29238 Brest, France5 Logistics Engineering College, Shanghai Maritime University, Shanghai 201306, China* Correspondence: Mohamed.Benbouzid@univ-brest.fr; Tel.: +33-2980-18007

Received: 1 April 2020; Accepted: 14 April 2020; Published: 16 April 2020

Abstract: The selection of electric machines for an Electric Vehicle (EV) is mainly based on reliability,efficiency, and robustness, which makes the 5-phase Permanent Magnet Synchronous Motor (PMSM)among the best candidates. However, control performance of any motor drive can be deeply affectedby both: (1) internal disturbances caused by parametric variations and model uncertainties and(2) external disturbances related to sensor faults or unexpected speed or torque variation. To ensurestability under those conditions, an Active Disturbance Rejection Controller (ADRC) based on anonline dynamic compensation of estimated internal and external disturbances, and a Linear ADRC(LADRC) are investigated in this paper. The control performance was compared with traditionalcontroller and evaluated by considering parametric variation, unmodeled disturbances, and speedsensor fault. The achieved results clearly highlight the effectiveness and high control performance ofthe proposed ADRC-based strategies.

Keywords: electric vehicle; 5-phase permanent magnetic synchronous motor; ADRC; LADRC; speedsensor failure

1. Introduction

Owing to its several advantages, 5-phase PMSM becomes over the years one of the best choices forthe electric vehicle and other applications which require robustness and efficiency. In addition to theclassical PMSM advantages such as long life, small size, simple structure, high torque to inertia ratio,easy control, etc., the high number of phases offer more robustness toward phase failures and reducesconsiderably the the torque ripples [1–3]. Nevertheless, the stability and precision of the control systemis highly affected by the system non-linearity, the load torque and parametric variations and externalperturbations [4]. Several strategies were introduced to improve the overall control performance byusing model linearization of the multi-phase PMSM and decomposing it into virtual sub-machinesmechanically coupled and magnetically decoupled. The developed control techniques can successfullystabilise the system under normal conditions but reached their limits in presence of unmodeled orunexpected disturbances. To overcome these limitations, the Active Disturbance Rejection Controller(ADRC) is investigated in this paper.

ADRC is a modern control strategy characterized by its real-time estimation and compensationfor internal and external disturbances [5,6]. It is composed mainly by a Tracking Differentiator

Machines 2020, 8, 17; doi:10.3390/machines8020017 www.mdpi.com/journal/machines

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(TD), an Extended State Observer (ESO), and Non-Linear State Error Feedback (NLSEF) control law.This technique is known by its simple structure, fast response, high precision, and easy parametersetting, beside its ability to deal with disturbances and uncertainties without the needs of an accuratemathematical model of the controlled system [7–9]. It also proved its worth in many applications, suchServo System of Self-Propelled Gun [10] , Energy Conservation Controller [11], Wind Turbine [12,13],Humanoid robot [14], and especially in the field of motor control [15–17].

For electric vehicle application, the ADRC technique was used in the speed control loop [18].The objective of this approach was to cancel the impact of unknown disturbances and parameteruncertainties on cruise control by compensating nonlinear vehicle dynamics. The proposed strategyproved its ability to overcome the effect of road slope and its capacity to improve stability and trackingperformance of the closed-loop system during load change. ADRC was also used to control the brakingcurrent by a real time estimation including the unmodeled perturbation of regenerative brakingsystem of EV [19]. This technique showed a high recovery efficiency against unknown disturbance.To improve the dynamic and steady state execution of the Brushless motor used in EV, authors in [20]implemented ADRC approach to control the speed loop. This technique has an important capacity toreduce torque ripple and enhance the overall performance of the controlled system under differentoperation conditions.

Over and above the modeling uncertainties and unknown perturbation, a failure in electricaldevices affects considerably the control performance of electric machines. A faulty mode is an irregularoperating condition, where the physical parameters of the system deviate from nominal value orstates [21]. There are several kinds of failures in electrical devices, among them, those related tosensors. Because of the importance of the sensors to get feedback information, it is unavoidable tofind out solutions that can ensure the continuity and the stability of the motor even in presence ofsensor failures. The core of the resilient control strategies proposed in literature is to find a techniqueable to switch between two different control strategies: one for the normal operation and the secondfor the faulty mode. This switching mechanism is based on the use of an observer to detect, isolate,and accommodate the sensor failures [22,23]. There are several study results dealing with the PMSMresilient control, mostly focused on an open-phase failure [24] and stator winding shorts [25] whileothers consider speed sensor sensor failure resilient control. Many of the proposed resilient controlstrategies are based on a detection and isolation mechanism [22,23,26–28].

This paper discusses the ADRC technique’S ability to control the 5-phase PMSM in case ofspeed and load torque variation while experiencing speed sensor failures. Thus, the ADRC approachis considered to be a resilient control algorithm without the need for the conventional switchingmechanism or an observer, usually used to estimate the failure and the system internal state. A standardADRC model has three main parts: TD, ESO, and NLSEF. This control approach is associated with adiscretization process, which increase the nonlinear ADRC adjustment complexity. For simplificationpurposes, the TD module was removed and the NLSEF function was replaced by a proportionalgain [29]. This modification consists of transforming the traditional form of the nonlinear ADRC toa linear one (LADRC) [12]. The LADRC mainly comprises the proportional controller and the ESOwhile keeping the same high performance of the nonlinear ADRC approach [30,31].

Our contribution in this paper is to investigate the efficiency of ADRC technique in controllingthe 5-phase PMSM for EV as a resilient control strategy, and under different operation conditions. Forthis purpose, this paper is structured as follows: the Section 2 presents a general model of the EVand the 5-phase PMSM. The Section 3 introduces the mathematical model of the Active DisturbanceRejection Controller and Linear ADRC used to simplify the motor control. The Section 4 presents thespeed and current control by the ADRC and LADRC, while in the Section 5, the simulation results ofthe 5-phase PMSM operation are presented and compared to PI regulator. The Section 6 is reserved forthe conclusion.

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2. Electric Vehicle Modeling

The electric vehicle model can be decomposed in four parts: the mechanical part, the electricmachine part, the controller part, and finally the observer part. Figure 1 shows the EV model blockdiagram using two ADRC strategies.

Figure 1. Block diagram of ADRC for a 5-phase PMSM.

2.1. Mechanical Part

In this study, the pitch and lateral movement of the EV are neglected and assume that the EV runsin horizontal straight line to simplify the dynamics model.According to aerodynamics principles, then the vehicle mechanics, the road load FT of the EV can beexpressed as follows [1]:

FT = Fr + Facc + Faero (1)

withFr = µmg (2)

Facc = ma = mdVdt

(3)

Faero =12

ρV2S f Cw (4)

The rolling resistance force represents the friction between the road and the wheels, named also thesolid friction force. This force depends usually on the type of vehicle, the road surface, the types of tiresand their pressure. The acceleration force, called inertia force is the force when the vehicle accelerate ordecelerate. The aerodynamic drag force called also the viscous resistance force is the force that opposesthe advancement of the vehicle in the air. In a vehicle, a gear is the mechanical part which connectsthe wheels and the motor. The relation of speed and torque between the motor and the wheels can bedescribed as follows:

ωw =ωm

ng(5)

andTL =

Tw

ηng=

rηng

FT (6)

Then the relation between the rotational motor speed and torque can be established by:

Jdωm

dt= Tem − f ωm − TL (7)

with TL is the total load torque applied to the motor.

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Since the load torque is a function of the forces applied to the EV as presented in Equation (6),it can be defined as an unknown parameter changing its value during the EV operation. This undefinedparameter is considered to be an external perturbation to the electric motor and directly affects thespeed dynamic according to Equation (7). Therefore, it is important to define a suitable control strategyable to compensate the external variation, and keep the high performance of the electric motor overthe unmodeled disturbance.

2.2. 5-Phase PMSM Modeling

To simplify the electric machine modelling, it is assumed that the 5-phases windings are regularlyshifted by 2π

5 and the saturation of the magnetic circuits and armature reaction are neglected [1].The mathematical model of the 5-phases PMSM in the rotating frame is described as follows [32]:

vdp = Ridp + Lpdidpdt − npωmLpiqp

vqp = Riqp + Lpdiqpdt + npωmLpidp +

√52 k1ωm

vds = Rids + Lsdidsdt − 3npωmLsiqs

vqs = Riqs + Lsdiqsdt + 3npωmLsids −

√52 k3ωm

Tem =√

52 (k1iqp − k3iqs)

J dωmdt = Tem − Bωm − TL

(8)

3. Active Disturbance Rejection Controller

The PMSM physical model can be affected by several internal and external perturbations,parametric variations, and potential sensor failures. Those perturbations will decrease the motorcontrol performance. Using ADRC, as a robust and resilient control strategy, those mixed uncertaintiesand failures are treated as a total disturbance, which is estimated and compensated in real time. Theparticularity of this control technique is its ability to estimate the whole disturbance as an extendedstate [33,34].

3.1. Resilient Control

Fault-tolerant or resilient control strategies should allow the controlled system, the EVpropulsion in our case; to be able accommodating failures while preserving stability conditions andmaintaining tracking performance as close as possible to the set-point, even in presence of unknownperturbations [35]. Several studies focused on resilient control techniques that consider sensor failuresproducing wrong feedback signals therefore perturbing the system control. Most of the proposedresilient control approaches use two control strategies, one devoted to the healthy state while switchingto another one when a sensor failure is detected [22].

In contrast with traditionally adopted resilient control approaches, the ADRC technique do notuse a separate observer or a switching mechanism, since it includes an extended observer, whichis able to estimate the internal state and the total disturbances. By defining the sensor failure as anexternal unknown perturbation, the ADRC is able to estimate and compensate its effect as a part of thetotal disturbance.

3.2. Nonlinear ADRC Design

The ADRC technique comprises three principal parts: a TD responsible for tracking signalsefficiently, the ESO which estimates the total disturbances and observe the internal state of the systemand the NLSEF or control law, used to provide a stable and effective output signal [36].

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3.2.1. Nonlinear Function Design

The nonlinear function is the key of the entire design in the ADRC approach. It is used in TD,ESO and NLSEF. The fal function, which is generally used, is described as follows:

f al(e, α, δ) =

|e|α sign(e), |e| > δ

eδ1−α , |e| ≤ δ

(9)

This equation can satisfy the conditions of continuity and derivability, where δ and α are two adjustableparameters, representing the filter and the nonlinear factor, respectively, with:

0 < α < 10 < δ

(10)

3.2.2. TD Design

Through the nonlinear function in ADRC, TD algorithm is able to arrange the transition processaccording to the reference input and achieve a smooth approximation of the generalized derivative ofthe input signal. Considering the first order system in a canonical form:

x(t) = f (x, t) + b.uy(t) = Cx(t)

(11)

In Equation (11) u is the known input, b is a constant, and the total disturbances f (x, t) is treated asthe external state variable to be estimated.

The first order system TD is described by [37]:ε0 = v1 − vv1 = −r f al(ε0, α0, δ0)

(12)

where v is the reference input, v1 is the trace output after TD arrangement and r is the trackingspeed factor.

3.2.3. ESO Design

The ESO estimates in real time the known, unknown, internal and external perturbations appliedto the system. The observed disturbances will be compensated, simultaneously as a feedback in orderto achieve object reconstruction and best performance. In this part, the ESO is introduced to estimatethe total disturbance f (x, t) and update the state variable in real time. Since the system to be controlledis a first order, then the observer must be designed as a second-order extended state observer ESO.The algorithmic expression of ESO can be designed as:

ε1 = z1 − yz1 = z2 + b0u− ρ1 f al(ε1, α1, δ1)

z2 = −ρ2 f al(ε1, α1, δ1)

(13)

where z1 and z2 are the output and the total disturbance estimations, respectively. e is the stateerror and b0 is an estimated value of the constant b. ρ1 and ρ2 are the correction gains of the outputerror factor.

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3.2.4. NLSEF Design

the NLSEF part is responsible to arrange the compensation of the control quantity and improvethe efficiency of the feedback control. It is described as the error between TD and ESO and expressedby the following equations:

ε2 = v1 − yu0 = ρ3 f al(ε2, α2, δ2)

u1 = u0−z2b0

(14)

where u0 and u1 are the objective control law and the final control signal after compensation,respectively. The different gains ρ1, ρ2 and ρ3 depend mainly on the control system sampling time.

3.3. Linear ADRC Design

As described, the ADRC controller requires the adjustment of a significant number of parameterssuch as: α, δ and ρ which complicate their tuning and may cause overshoot and hyper harmonicnoise [5]. To simplify the controller, some researchers propose a Linear ADRC (LADRC) toreduce perturbations and non-linearity in the control system, and to drive the PMSM with morerobustness [30,31]. In LADRC approach, the linear TD (LTD) is expressed by:

ε = v1 − vv1 = −rε

(15)

and the Linear ESO (LESO) can be defined as follows:ε = z1 − x1

z1 = z2 + b0u− ρ1ε

z2 = −ρ2ε

(16)

where ε is the estimated error, z1, z2 and x1 are the estimated state variable, the estimation of totaldisturbance and the actual state variable, respectively. TD and LTD are used to smooth the input andreduce the overshoot, but it may extend the system’s adjustment time. For this reason, taking intoaccount adjustment time and overshoot, requires to take a compromise solution. Moreover, to simplifythe system structure, NLSEF, used in ADRC, can be replaced by a simple proportional controller anddefined as a control law of LADRC [12,38].

This control law is presented by:ε = v− yu0 = kpε = kp(v− y)u1 = u0−z2

b0

(17)

Then, the LADRC is composed mainly by a proportional controller and LESO. The LESO is usedto estimate the total disturbances which include internal and external ones as well as the uncertaintiesand the system states. For the 5-phase PMSM, five controllers are required, where four are used tocontrol the currents of primary and secondary dq frames and one for the speed loop.

3.4. Stability Analysis

Stability study of a closed-loop linear control system is usually carried out by the directdetermination of its own values. However, for a nonlinear control system, the use of a Lyapunovfunction is the most popular technique for stability analysis. In [39], the stability study of the ADRCregulator and ESO observer was demonstrated using a Lyapunov function. Similarly for the linearADRC regulators presented in [34,40], the analysis of its stability based on the Lyapunov functionjustifies the existence of appropriate gains ensuring estimation errors convergence.

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Firstly, based on the linear ADRC study presented in [38], the nonlinear system defined byEquation (11) can be rewritten as:

x1 = x2 + bux2 = hy = x1

where h = f (18)

Then, the matrix form of Equation (18) is:x(t) = Ax + bBu + Ehy(t) = Cx(t)

(19)

with A =

[0 10 0

], B =

[10

], E =

[01

], and C =

[1 0

]The LESO presented by Equation (16) can then be deduced and described by the

following expression: z = Az + b0Bu− Lε2

y = −Cz(20)

with L =

[ρ1

ρ2

]=

[2ω0

ω20

]; L denote the LESO gains vector and ω0 its bandwidth. For the stability

analysis, let define the dynamic of the tracking error e = x− z of the observer by the following function:

e = Aee + d (21)

with Ae = A− βC =

[−ρ1 1−ρ2 0

]and d = Eh.

Lemma 1. It can be noted that for any bounded h, the error e is bounded if the matrix Ae is Hurwitz.

Proof. Assuming the matrix Ae is Hurwitz, let define respectively the Lyapunov function V and theLyapunov equation by:

V = eTXe (22)

andAT

e X + XAe = −P (23)

where X is the unique solution of the Lyapunov equation and P is a definite positive matrix.According to [41], the derivative of Lyapunov function is expressed in the form:

V = −eT Pe + 2dTXe= −(eT P

12 − dTXP

12 )(eT P

12 − dTXP

12 )T + (dTXP

12 )(dTXP

12 )T (24)

which implies that V < 0 if: ∥∥∥eT P12 − dTXP

12

∥∥∥2>∥∥∥dTXP

12

∥∥∥2

(25)

or equivalently ∥∥∥eT P12

∥∥∥2> 2

∥∥∥dTXP12

∥∥∥2

(26)

For P = I, an identity matrix, the derivative of Lyapunov function V defined by Equation (24) isnegative if:

‖e‖2 > 2 ‖Xd‖2 (27)

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which implies for all error e satisfying Equation (27), ‖e‖2 decreases. Then, it can be noted that eis bounded.

Lemma 1 can be generalized for any system described by:

µ = Nµ + f (µ) (28)

with µ ∈ <n and N ∈ <nxn. The corresponding lemma is:

Lemma 2. If the matrix N is Hurwitz and the function f (µ) is bounded, then the state µ in Equation (28) isalso bounded.

Combining Lemmas 1 and 2, the boundedness of LADRC can be defined by [41]:

Theorem 1. If the control law Equation (17) and the observer Equation (20) are stable, the linear ADRC designof Equations (20) and (17) presents a stable closed-loop system with bounded input and output.

The above-presented stability analysis shows the ability of the ADRC controller to solve theuncertainties problem in a real system.

4. ADRC Design for 5-Phase PMSM

Based on the above analysis, the ADRC technique can estimate and compensate the internal andexternal disturbances of the system in real time and independently of the mathematical model of thesystem [15].

4.1. Speed Controller Design

The ADRC technique is based on the creation of an extended model of order n + 1, where n is theorder for the system to be controlled [42]. The additional state variable, called total disturbances, mustbe properly defined and includes the nonlinear terms and all kinds of disturbances .

Based on the 5-phase PMSM model, the following equation can be obtained:

ωm =T∗em

J− B

Jωm −

TLJ

=

√52(

k1

Ji∗qp −

k3

Ji∗qs)−

BJ

ωm −TLJ

(29)

It can be noted from Equation (29) that the moment of inertia J, friction coefficient B and thevariations of the load torque TL could affect the control precision of the system. For this reason, themechanical Equation (29) can be modified as:

ωm =T∗em

J+ f1 (30)

For the design of the speed controller, the variable to be controlled is the rotor speed y = ωm, andthe controller output is the electromagnetic torque, which is a function of q-axis currents, u = Tem.Then, we can consider that the q-axis currents are the indirect outputs of the controller system, andall other factors are considered to be system disturbances. The block diagram of the ADRC speedcontroller for the 5-phase PMSM can be structured as described in Figure 2

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Figure 2. Schematic diagram of standard ADRC controller.

where v1 is the tracking signal of the speed reference ω∗m, ε1 describe the error signal of speedloop. u0 is the output of the NLSEF and u is the referential torque compensated by the estimateddisturbance. z1 and z2 represent the tracking signal of ωm and the estimated disturbance, respectively.

4.2. ADRC Controller Current Loop

In this section, we apply an ADRC to the current system of PMSM to achieve the robust controlsystem design. Based on the mathematical model of PMSM, the state equation of current loop can bedescribed by:

x(t) = Ax(t) + Bu(t) + f2(x(t))y(t) = x(t)

(31)

with u = [vdp, vqp, vds, vqs]T is the control input, x = [idp, iqp, ids, iqs]T is the state variable, y is thesystem’s output, f2 represent the nonlinear terms.

A = −

R/

Lp 0 0 00 R

/Lp 0 0

0 0 R/Ls 00 0 0 R/Ls

B =

1/

Lp 0 0 00 1

/Lp 0 0

0 0 1/Ls 00 0 0 1/Ls

f2(x) =

npωmiqp

−npωmidp −√

5/2ωmk1/

Lp

3npωmiqs

−3npωmids +√

5/2ωmk3/

Ls

Following the same approach as for the speed control loop, the state Equation (31) can be expressed

as follows: x(t) = Bu(t) + f3(x, t)y(t) = x(t)

(32)

where f3 represents the total disturbances which includes non-linearity, external and internaldisturbances and all other terms other than the control inputs.

5. Simulation Results

The ADRC control performance is compared simultaneously with that of LADRC and theconventional PI regulator to show the properties of each regulator and their capabilities. The Simulationwere carried out in Matlab/Simulink platform. The PMSM and EV numerical parameters are presentedin Tables 1 and 2.

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Table 1. 5-phase PMSM Parameters.

Symbol Parameter Value

np Number of pole pairs 2R Stator resistance 5 ΩLp Principal machine inductance 0.1228 HLs Seconder machine inductance 0.0222 Hk1 First harmonic constants 2k3 Third harmonic constants 0.66J Inertia moment 0.00075 kg·m2

B Viscous friction coefficient 0.000457 N.m.s/rad

Table 2. EV Parameters.

Symbol Parameter Value

µ Rolling resistance coefficient 0.015m Mass of the EV 1000 kgρ Air density 1.2 kg/m3

S f Frontal area 2.5 m2

Cw Drag coefficient 0.3g Gravity acceleration 9.81 m/s2

r Tyre radius 0.3 m

For comparison purposes, the following fitness functions are defined to study the regulatorsperformance [43]:

ISE =∫

e(t)2.dt (33)

IAE =∫|e(t)| .dt (34)

ITAE =∫

t. |e(t)| .dt (35)

ITSE =∫

t.∣∣∣e(t)2

∣∣∣ .dt (36)

Equations (33)–(36) present the Integral of the Square Error (ISE), the Integral of Absolute Error (IAE),the Integral of the Time-weighted Absolute Error (ITAE), and the Integral of the Time Square Error(ITSE), respectively [44].

As already announced, the control of PMSM requires the use of five control block for both ofcurrents and rotor speed, with an adjustment of a significant number of gains. Then, it is important toanalysis the effect of variation of control parameters on PMSM operation. As presented in Equation (9),nonlinear law used in ADRC depends mainly on α and δ. For 0 < α < 1, the time response increaseand the signal track rapidly its reference to the increasing of gain α which improve the control efficiencyas shown in Figure 3a. Also, the control law depends on δ variation, where the time response increasewith decreasing of δ as presented in Figure 3b.

(a) α0 gain variation (b) δ0 gain variationFigure 3. Control law variation.

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To overcome the complexity of chosing the adequate gains, the LADRC was proposed to minimizethe number of parameters to be adjusted and to simplify the control approach.

5.1. Speed and Load Torque Variation

Figures 4 and 5 present the respectively the speed, electromagnetic torque and the q-axes currentsresponses during the motor operation. Firstly, the speed was maintained to 1500 RPM with a variabletorque, which progressively increases until it reaches its nominal value, as is shown in Figure 4. In thesecond case, the load torque was set to be constant with a speed variation, as presented in Figure 5.

(a) Rotor speed (b) Torque

(c) currents of the primary dq frame (d) Currents of the secondary dq frameFigure 4. PMSM performance with load torque variation.

(a) Rotor speed (b) Torque

(c) Currents of the primary dq frame (d) Currents of the secondary dq frameFigure 5. PMSM performance with speed variation.

The three regulators, ADRC, LADRC and the PI show a high tracking performance where thephysical parameters track perfectly its references, especially in the steady state. The difference betweenthe three regulators appears in the transient state. In the case of Pi regulator, a reduction in responsetime is usually accompanied by a significant overshoot, whereas the resolution of this overshootproduces a longer response time or even a delay in the system, which is not the case for the ADRC andLADRC regulators as shown in the following simulation. The ADRC regulator presents a response timethat is very close to that of PI, but with a reduced overshoot. By simplifying the structure of the ADRC

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and minimizing regulation parameters, the LADRC combine between the both characteristics: thestability and the faster response. It was totally remarkable that LADRC presents a better performancethan the standard ADRC and the traditional PI regulator, with a faster speed response as presented inboth of Figures 4 and 5.

Also, we notice in Figures 4c,d and 5c,d, that both of the ADRC and LADRC regulate the q axiscurrents to track their references so they can adapt perfectly to those variations. The performance ofeach regulator is given Tables 3–5.

Table 3. Performance indices of the speed control loop.

Starting Stage Torque Disturbance Speed Variation

LADRC IAE 0.0207 1.3897 0.4374ISE 0.0072 30.645 1.1128

ITAE 0.0043 1.0190 0.2112ITSE 0.0013 22.003 0.5233

ADRC IAE 2.2360 1.4314 24.242ISE 54.016 32.315 3406.7

ITAE 0.4780 1.0497 12.068ITSE 10.325 23.205 1655.4

PI IAE 1.4552 6.8994 15.557ISE 19.674 1065.1 1521.2

ITAE 0.3210 5.0133 7.5860ITSE 3.8406 763.27 720.50

Table 4. Performance indices of the current control loop.

Starting Stage Torque Disturbance Speed Variation

LADRC IAE 0.0188 0.1280 0.0053ISE 0.0060 0.0802 0.00017

ITAE 0.00067 0.0770 0.0024ITSE 0.00011 0.0481 0.00007

ADRC IAE 0.0113 0.1280 0.0037ISE 0.0017 0.0802 0.00004

ITAE 0.00075 0.0770 0.0018ITSE 0.00003 0.0481 0.00002

PI IAE 0.0472 0.1367 0.0115ISE 0.0160 0.0821 0.00062

ITAE 0.0037 0.0834 0.0062ITSE 0.0008 0.0495 0.00033

Table 5. Performance indices of iqs in different operating conditions.

Starting Stage Torque Disturbance Speed Variation

LADRC IAE 0.0562 0.1930 0.0332ISE 0.0509 0.1811 0.0080

ITAE 0.0021 0.1162 0.0156ITSE 0.0010 0.1087 0.0037

ADRC IAE 0.0957 0.1931 0.0329ISE 0.0642 0.1811 0.0054

ITAE 0.0069 0.1163 0.0170ITSE 0.0031 0.1087 0.0027

PI IAE 0.1942 0.2137 0.0637ISE 0.2933 0.1896 0.0212

ITAE 0.0139 0.1317 0.0329ITSE 0.0137 0.1148 0.0107

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Performance indices of speed, iqp, and iqs are compared in case of starting stage, load torqueand speed variations. The lowest performance indices for IAE, ISE, ITAE, and ITSE values prove theefficiency of proposed control strategy to achieve the highest tracking performance. These simulationresults clearly show that the best control performance is achieved with ADRC and LADRC.

5.2. Speed Sensor Failure

In this section, the proposed control strategies are tested under speed sensor failure conditions. Inthis context, Figure 6 illustrates the additive signals that describe the considered sensor failures. Thefirst signal, inserted to the rotor speed sensor output, represents a simple step with an amplitude equalto 150 RPM (Figure 6a), and the second one constitute an unmodeled noise with an amplitude below6% of the nominal speed.

Figures 7 and 8 illustrate the PMSP rotor speed, torque, and currents when considering speedsensor failures.

(a) Step error (b) Unmodeled noise errorFigure 6. Failures of speed sensor.

(a) Rotor speed (b) Torque

(c) Currents of the primary dq frame (d) Currents of the secondary dq frameFigure 7. PMSM performance in the first faulty mode.

In the obtained simulation results, it can be noted that the 5-phase PMSM achieves very interestingtracking performance in terms of speed and torque despite the speed sensor failure. Indeed, the 5-phasePMSM keeps its speed and produced electromagnetic torque too smooth, as in the healthy mode, andrapidly compensate the difference between the reference signal and the faulty one. Using the ESO,included in the ADRC controller, the sensor failure can be estimated in real time as a part of the totaldisturbance affecting the motor, and compensated by the control law to ensure the system prescribedperformance. This mechanism justifies the choice of the ADRC controller as a resilient control strategy,in addition to its robustness ability. These simulations results clearly highlight the robustness andresilience control performance of the 5-phase PMSM thanks to the use of the ADRC and LADRC.

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(a) Rotor speed (b) Torque

(c) Currents of the primary dq frame (d) Currents of the secondary dq frameFigure 8. PMSM performance in the second faulty mode.

5.3. Simulation Results Using Driving Cycles

To test the validity of this control strategy, the proposed system operation will be tested with theWorldwide Harmonized Light Vehicle Test Procedure (WLTP) developed by the European Union [45].This developed driving cycle is based on real-driving data, gathered from around the world [45,46].As shown in Figure 9, WLTP is composed of four different parts with various average speeds: low,medium, high and extra high. Each one of those parts contains a variety of driving phases, stops,acceleration and braking phases.

(a) Electric vehicle speed (b) Torque

(c) Currents of the primary dq frame (d) Currents of the secondary dq frameFigure 9. PMSM performance under WLTP drive cycle.

It can be noted from the simulation results that because of the TD arrangement and the NLSEF,the PMSM can start continuously without overshoot but with the slow speed response.

In case of LADRC with elimination of TD, the system presents a neglected overshoot in a transientstate, but it reaches quickly their reference value and have great stability in the steady state. Thesimulation results demonstrated the efficiency of suggested control strategy and the characteristics ofeach one in terms of fast tracking and robustness for internal and external disturbances

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6. Conclusion

This paper dealt with the active disturbance rejection control of a 5-Phase PMSM-based electricvehicle. In this context, the tracking capability and the resilient control performance of the proposedADRC technique were investigated in different operation conditions and compared to its simplifiedstructure presented by the linear ADRC and to the traditional PI controller. Standard forms of ADRCand Linear ADRC were implemented to control the speed and current considering rotor speed andload torque variations. Despite the difference between the two models, they have the same operatingprinciple. Based on estimation and compensation of external and internal disturbances, they ensuredhigher performance and robustness compared to the PI controller. In faulty speed sensor conditions,the failure was estimated by the ESO as a part of the total disturbance and compensated in the real-timeusing the control law. According to the achieved simulation results, it was shown that the linear ARDCseems to be the most promising control approach for real-world application implementation as itallows leading to better control performance compared to ADRC and PI controllers.

Author Contributions: Conceptualization, A.H., S.B.E., Z.Z., and M.B.; Methodology, A.H., S.B., Z.Z., and M.B.;Software, A.H.; Validation, A.H., S.B., and Y.B.; Formal Analysis, A.H., S.B.E., and M.B.; Investigation, A.H.;Writing—Original Draft Preparation, A.H.; Writing—Review & Editing, A.H., S.B.E., Y.B., Z.Z., M.B., and M.N.A.;Supervision, M.N.A. All authors have read and agreed to the published version of the manuscript.

Funding: This work received no funding.

Acknowledgments: This work was supported by the Ministry of Higher Education and Scientific Researchof Tunisia.

Conflicts of Interest: The authors declare no conflict of interest.

Nomenclature

p,s primary and secondary fictitiousmachine, respectively, in the dq frame.,

LADRC Linear Active Disturbance RejectionController.

v Stator voltage, Tw Torque of wheel.i Stator currents, TL Torque of motor.Tem Electromagnetic torque, η Efficiency coefficient.L Virtual machine inductance, ng Gear ratio.V Driving velocity, EV Electric Vehicle.a Vehicle acceleration, TD Tracking Differentiator.Fr Rolling resistance force, ESO Extended State Observer.Facc Acceleration force, NLSEF Non-Linear State Error Feedback.Faero Aerodynamic drag force, PMSM Permanent Magnet Synchronous Motor.ωw Rotational speed of wheel, ADRC Active Disturbance Rejection Controller.ωm Rotational speed of motor,

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